374 



SURFACE WAVES. 



[CHAP, ix 



wards ; thus at a depth of a wave-length the diminution of 

 amplitude is in the ratio e~ 2n or 1/535. The forms of the lines 

 of (oscillatory) motion (-vjr = const.), for this case, are shewn 

 in the annexed figure. 



In the above investigation the fluid is supposed to extend to infinity in 

 the direction of x, and there is consequently no restriction to the value of k. 

 The formulae also, give, however, the longitudinal oscillations in a canal of finite 

 length, provided k have the proper values. If the fluid be bounded by the 

 vertical planes x 0, x = I (say ), the condition d(f)/dx = is satisfied at both ends 

 provided sin^ = 0, or kl = mn, where m = l, 2, 3, .... The wave-lengths of the 

 normal modes are therefore given by the formula \ = 2l/m, Cf. Art. 175. 



218. The investigation of the preceding Art. relates to the 

 case of standing waves ; it naturally claims the first place, as a 

 straightforward application of the usual method of treating the 

 free oscillations of a system about a state of equilibrium. 



In the case, however, of a sheet of water, or a canal, of uniform 

 depth, extending to infinity in both directions, we can, by super 

 position of two systems of standing waves of the same wave-length, 

 obtain a system of progressive waves which advance unchanged 

 with constant velocity. For this, it is necessary that the crests 

 and troughs of one component system should coincide (horizon 

 tally) with the nodes of the other, that the amplitudes of the two 

 systems should be equal, and that their phases should differ by a 

 quarter-period. 



Thus if we put v Vi v-2 (1), 



where % = a sin kx cos at, 7j 2 = a cos kx sin at (2), 



we get 97 = a sin (kx a-t} (3), 



which represents (Art. 167) an infinite train of waves travelling 



